Keeping the image sharp depends a lot on the image and what you are trying to do with it. Resizing screen captures has always been and will always be problematic because there are a bunch of single pixel wide lines in most web pages and most software UI designs. A screenshot of the problem footage and some project details may help us help you with your project.
On the better resizing algorithms, they are often requested and improvements have been made and will continue to be developed. For some projects I have recently started using Camera Raw on import to clean up and improve problematic footage. This may help your project also, but without details there's not much I can do to suggest a solution.
If you want more resizing options do what I have done. Fill out a feature request. The process of adding features is surpriseingly democratic. The request with the most votes gets the most attention.
RevisionFX' RE:Map has an alternate corner pin with tweakable interpolation options and then there is Red Giant's 3D Warp thingy. Not sure if they still sell it. +1 on the interpolation. Being stuck with bilinera and a bicubic filter that is just as bad as Photoshop's certainly makes look AE look very bad...
Thanks Mylenium, the sharpen controls on the RE:Map Planar plugin do help.
I'm amazed resizing filters haven't been added yet. It seems that in the last few years Adobe are determined to keep on adding stuff to AE to make it do more stuff than making it better at doing the stuff it already does.
Camera tracker - there is much better software for it than that and it cheapens the art of matchmoving.
Mask tracker - AE comes with Mocha.
Cineware - Painfully slow to use.
I would much rather have proper filters, GPU accelerated effects and disk playback as opposed to any of that stuff.
The brief was to take out the white line. Doing screen inserts isn't much of an issue as generally the CP is resizing down. However for this I was using Mochas option to push the surface to the corners of the image. This means you've got the CP affecting the image at 100% and sometimes having to scale up from that as well.
I'm amazed resizing filters haven't been added yet.
Well, I'm sure that is on everyone's mind, but since it's "core" stuff that affects pretty much every other feature, it may be difficult to fix without breaking other stuff...